Apologizing and being sorry are quite different, yet used as if they were the same.
There are lots of ways to respond these days (in person, email, phone, text) and there are generally two phrases used in setting the situation right in your mind.
I’ve seen both and have to say, there is a very notable difference (to me) when someone takes the time to say “I’m sorry” versus someone that says “I apologize”. Maybe I’m just being oversensitive to this one, but seriously, it is a lot harder to say that you are sorry and mean what you are saying than it is to “apologize”.
The delivery vehicle is just as important and highlights the good or bad of the words chosen.
Walking up to a person or calling them on the phone and saying to them “I’m sorry that what I did cause problems for you…” takes a lot more effort and sincerity than sending an email or text that reads, “My apologies for any inconvenience that causes…”.
Be mindful of your words and choice of delivery.